\chapter{Pattern forming nonlinear optics} % (fold)
\label{cha:patterns}

Transverse optical patterns are fundamental to the operation of the all-optical switch I present in this thesis. This chapter serves to outline the minimum ingredients needed to observe transverse patterns in a nonlinear optical system. 

The primary requirement for transverse pattern formation is a mechanism that provides gain for the portion of a wave that propagates with an off-axis wavevector. If such off-axis gain is large enough, the stable state of plane-wave, or gaussian-beam propagation can become unstable and exhibit growth of off-axis components. These off-axis components results in transverse field structure that can be observed as an intensity pattern in the far field. An example pattern, shown in Fig.~\ref{fig:hexagons}, is the set of six spots that I typically observe in my experiment. Since the patterns I use for all-optical switching exhibit hexagonal symmetry, the ultimate goal of this chapter is to demonstrate the necessary ingredients for a model that exhibits hexagonal pattern formation.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/example_pattern.pdf}
  \end{center}
  \caption[Example pattern formed by a counterpropagating beam system]{Beams counterpropagating through a nonlinear medium give rise to transverse structure in the field in the plane perpendicular to the direction of propagation. a) Beams and nonlinear medium. b) Hexagonal pattern. c) Pump beam transmitted off-resonance (for reference)}
  \label{fig:hexagons}
\end{figure}

Many optical systems exhibit pattern formation through various mechanisms. As an example, a single laser beam propagating through a gas of sodium atoms will exhibit transverse pattern formation if the beam intensity is large enough \cite{Bennink_2002aa}. However, for a pattern-forming system to be applicable to ultra-low-light all-optical switching, the power required for pattern generation should be minimized. The use of two beams in the counterpropagating geometry serves to reduce the intensity required for the generation of transverse patterns \cite{Gaeta_1993aa}, as will be shown near the end of this chapter. Hence, two beams counterpropagating through a nonlinear medium form the specific class of pattern forming systems that are the focus of the following sections. 

Throughout the past thirty years there have been several dozen experimental investigations of counterpropagating beam instabilities that explore the system's rich nonlinear nature and large parameter space (see Ref.~\cite{Gaeta_1993aa} and references therein). Although the large number of possible beam configurations and experimental conditions have only been partially investigated, the observed instabilities exhibit several generic features such as periodic and chaotic power fluctuations, polarization and amplitude bistability, and off-axis pattern generation.

Different theoretical treatments account for each of these features independently but few theories explain all of them simultaneously. The primary difficulty lies in describing the details of the nonlinear response to the applied field. A reasonable approximation is to treat the medium as a Kerr medium, with an intensity-dependent refractive index $n=n_0+n_2I$. The Kerr model, by including only intensity-dependent terms, ignores nonlinearities above third-order. It also assumes a transparent medium, \emph{i.e.}, one with no absorption. Despite these approximations, this model has been very successful at describing a wide range of nonlinear phenomena \cite{Boyd_2002aa}. In certain cases, such as near an atomic resonance, other models of the nonlinear medium have better quantitative accuracy, but often exhibit qualitative features that are similar to those described by Kerr nonlinearity. Finally, the Kerr model is well studied and, as the following sections show, it includes sufficient details of the nonlinear optical interaction to account for the generation of transverse optical patterns in a counterpropagating-beam system.

In a Kerr medium, an instability that occurs on the pump-beam axis can be described by a one dimensional theory (along the pump beam direction). However, for instabilities that generate light off-axis, accurate descriptions require the inclusion of transverse dimensions, or, at a minimum, one beam with a transverse component. A further complication is that many of the standard nonlinear optics approximations are not valid when the frequency of an optical field is tuned close to an atomic resonance, or when the vector nature of the field significantly changes the nonlinear interaction. For atomic vapors, the nonlinear response is due to optical pumping. When the incident intensity is sufficient to transfer population among the atomic states, the level populations must be included in a model for it to be quantitatively accurate. Some of these issues have been addressed in the various theoretical treatments that describe features of counterpropagating beam instabilities and the text below summarizes several important treatments that successfully capture the important features of counterpropagating beam systems.

Considering on-axis instabilities in the scalar approximation, where the vector nature of the electric field is neglected and the nonlinear atomic polarization is assumed to follow the form $P^{\mathrm{NL}}=\epsilon_0\chi^{(3)} E^3$, Silberberg and Bar-Joseph \cite{Silberberg_1984aa} found that counterpropagating beams become unstable, exhibiting periodic and chaotic intensity fluctuations.\footnote{Here, $\epsilon_0$ is the permittivity of free space, $\chi^{(3)}$ is the third-order nonlinear susceptibility and $E$ is the electric field amplitude, all in SI units.} Extensions of this result by Kaplan and Law \cite{Kaplan_1985aa} that allow for two field polarizations show on-axis polarization bistability that has since been experimentally verified \cite{Gauthier_1990aa}.

Off-axis generated light is described in an early treatment of counterpropagating beams by Yariv and Pepper \cite{Yariv_1977aa} where they propose an interaction now known as mirrorless parametric oscillation in which the gain in a parametric process diverges. Infinite gain indicates that vacuum fluctuations in the field can seed the generation of new beams of light. Depending on phase-matching conditions, these beams can emerge at an angle $\theta$ to the pump-beam axis, making this interaction one possible mechanism for generating off-axis patterns \cite{Grynberg_1988aa}. However, the generated light described by mirrorless parametric oscillation in \cite{Yariv_1977aa} is in the same state of polarization as the pump beams, which is not the case for all experimental results (see \cite{Maitre_1995aa} and references therein).

To account for the generation of cross-polarized fields with transverse wavevector components, a more complete theoretical treatment must be used. Gaeta and Boyd \cite{Gaeta_1993aa} consider the tensor properties of the nonlinear interaction and specify the thresholds for both amplitude and polarization instabilities with wavevector components in the transverse plane.

Finally, it should be noted that the majority of the stability analyses appearing in the literature are for equal-amplitude pump waves. For imbalanced pump waves, oscillatory instabilities can occur in addition to exponential-growth instabilities \cite{Gaeta_1993aa}, which would suggest that unbalanced pump beams reproduce qualitatively similar results with quantitatively different instability thresholds. Additionally, off-axis cross-polarized generated light is expected and experimentally confirmed for unbalanced pump beam powers \cite{Gauthier_1990aa,Dawes_2005aa,Dawes_2008aa}.

In the remaining sections of this chapter, I first describe a simple estimate for the angle $\theta$ between pump beams and generated off-axis beams. This estimate is based on the phenomena of weak-wave retardation that was discussed in Sec.~\ref{sec:all_optical_switching_via_nonlinear_phase_shift}. Following this estimate, I present three treatments, each of which includes new physics and describes an additional aspect of transverse pattern formation. I begin with the treatment of Yariv and Pepper and introduce the physical process known as backward four-wave mixing (BFWM) which can be used to describe gain experienced by an off-axis beam. Following this, I introduce forward four-wave mixing (FFWM) and then review the treatment by Grynberg and Paye that includes both FFWM and BFWM effects to describe the finite pattern angle observed in counterpropagating beam systems (see Fig.~\ref{fig:hexagons}). Finally, I discuss the model considered by Firth and Par\'e in which FFWM, BFWM, cross-coupling and transverse dimensions are simultaneously considered. Although this model remains simple by assuming scalar fields, an instantaneous nonlinear response, and no absorption, it demonstrates that transverse effects and counterpropagation are sufficient to result in spatial and temporal instabilities. Furthermore, as I show in Chapter~\ref{cha:nummodel}, this model exhibits patterns and switching behavior that are qualitatively similar to those I observe experimentally.

\section{Weak-wave retardation} % (fold)
\label{sec:weak_wave_retardation}

From the discussion in Sec.~\ref{sec:all_optical_switching_via_nonlinear_phase_shift}, we are familiar with the idea of weak-wave retardation. Originally discussed by Chiao \cite{Chiao_1966aa}, weak-wave retardation describes the phenomena that the nonlinear index of refraction experienced by a weak wave due to a strong wave is twice as large as the the nonlinear index of refraction experienced by a single strong wave due to the self nonlinear phase shift. The origin of this effect is the degeneracy factor in the description of the electric polarization of the material, and hence the two beams must be distinguishable either in frequency or propagation direction (or both). Because a strong wave and a weak wave experience different indices of refraction, they will necessarily have different propagation vectors 
\begin{equation}
  k_{w,s}=n_{w,s}\omega/c,
\end{equation}
where
\begin{align}
  n_w&=n_0+2n_2I\\ 
  n_s&=n_0+n_2I,
\end{align}
(for the derivation, see Sec.~\ref{sec:all_optical_switching_via_nonlinear_phase_shift}).
For a process involving weak waves, such as those generated by a pattern forming instability, and strong waves, such as the pump beams driving the system, weak-wave retardation provides the following simple argument for why the generated beams propagate at an angle to the pump beams.\footnote{It should be noted that I am not assuming anything about the mechanism of this instability, simply that some physical process involving forward-moving beams gives rise to weak fields and requires the generated fields be phase-matched in order for the process to occur efficiently. Specifically this process is known as forward four-wave mixing and is discussed in more detail in Sec.~\ref{sec:forward_four_wave_mixing}.}

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/wwr.pdf}
  \end{center}
  \caption[Weak-wave retardation in FFWM]{Weak-wave retardation provides an estimate for the phase-matching angle $\theta$ for a forward process involving a strong wave with wavevector $k_s$ and a weak wave with wavevector $k_w$. a) For $n_2>0$, $k_w$ is longer than $k_s$. b) $k_w$ and $k_s$ propagate at an angle $\theta$. c) For $n_2<0$, $k_w$ is shorter than $k_s$, and d) the process cannot be phase matched for any angle.}
  \label{fig:weak_waves}
\end{figure}

Consider two waves, one weak and one strong, with wavevectors $k_w$, and $k_s$ respectively. If we assume the weak wave is generated by a wave-mixing process that requires it to be phase matched with a strong pump wave propagating along the $z$-axis, then their projections along the $z$-axis must be equal. This is illustrated in Fig.~\ref{fig:weak_waves}. If the nonlinear refractive index is positive ($n_2>0$) the weak wave will have a longer $k$-vector due to the larger nonlinear refractive index and must propagate at an angle $\theta$ to the pump-beam axis $z$. We now estimate $\theta$ for a system that exhibits a nonlinear phase shift $k_s^\mathrm{NL}L=\pi/2$, where I define $k_s=k_0+k_s^\mathrm{NL}$ as a sum of linear and nonlinear contributions to the wavevector. Recall from Sec.~\ref{sec:all_optical_switching_via_nonlinear_phase_shift} that a relative phase shift of $\phi=\pi/2$ is required for an all-optical switch based on the nonlinear phase shift.

From geometrical arguments based on Fig.~\ref{fig:weak_waves}, we begin with
\begin{equation}
  \cos\theta=\frac{k_s}{k_w}.
\end{equation}
Separating the linear and nonlinear contributions to the wavevector gives
\begin{equation}
  \cos\theta=\frac{k_0+k_s^\mathrm{NL}}{k_0+k_w^\mathrm{NL}},
\end{equation}
where the nonlinear phase shift experienced by the strong wave corresponds to  $k_s^\mathrm{NL}=\pi/2L$. Weak wave retardation thus implies the phase shift experienced by the weak wave is twice a large, hence $k_w^\mathrm{NL}=\pi/L$. Finally, from $k_0=n_0 \omega/c\simeq2\pi/\lambda$, where I have approximated\footnote{The approximation $n_0\simeq1$ is valid for a dilute vapor. If the nonlinear medium were a transparent solid, $n_0$ would have a value greater than unity.} $n_0\simeq1$, we have
\begin{equation}
\cos\theta=\frac{\frac{2\pi}{\lambda}+\frac{\pi}{2L}}{\frac{2\pi}{\lambda}+\frac{\pi}{L}}=\frac{1+\frac{\lambda}{4L}}{1+\frac{\lambda}{2L}}\simeq 1-\frac{\lambda}{4L},
\end{equation}
where I have assumed that $\lambda<<L$. The small angle approximation is valid in this case, and thus
\begin{align}
  \label{eqn:estimate_theta}
  1-\frac{\theta^2}{2}&\simeq1-\frac{\lambda}{4L}\nonumber\\
  \theta&\simeq\sqrt{\frac{\lambda}{2L}}.
\end{align}
If I substitute my experimental values, $\lambda=780$~nm and $L=5$~cm, into Eq.~(\ref{eqn:estimate_theta}), I find that $\theta\simeq2.8$~mrad. This agrees qualitatively with what is measured in experiments (typically in the range of 2-4~mrad). Weak-wave retardation thus provides a good estimate for the origin of off-axis effects in wave mixing processes. The following sections outline such processes in more detail, and demonstrate their specific roles in the formation of transverse patterns.

% section weak_wave_retardation (end)

\section{Backward four-wave mixing} % (fold)
\label{sec:backward_four_wave_mixing}

Backward four-wave mixing in a nonlinear medium, illustrated in Fig.~\ref{fig:bfwm}, can be described in terms of a diffraction grating induced by the interference between two beams. In the case illustrated, a forward (right-moving) probe wave with amplitude $A_3$ interferes with a backward-moving pump wave, amplitude $A_2$, propagating at an angle $\theta$ to the $z$-axis. The intensity pattern due to this interference gives rise to a spatial variation of the index of refraction via the nonlinear index of refraction $n=n_0+n_2I$. A second pump wave, propagating in the forward direction, scatters off the index grating, into the direction opposite the incident probe.

\begin{figure}[htbp]
  \centering
    \includegraphics[scale=1]{Figures/four_wave_mixing.pdf}
  \caption[Backward four-wave mixing in a nonlinear medium]{Backward four-wave mixing in a nonlinear medium. a) An intensity grating established by the interference of one pump wave, $A_2$ incident from the right, and the signal wave, $A_3$ incident from the left, creates a refractive index grating. b) The second pump wave $A_1$, incident from the left, scatters off the index grating to provide the conjugate beam $A_4$, which exits the medium to the left.}
  \label{fig:bfwm}
\end{figure}

To examine the origins of gain in backward four-wave mixing (BFWM), consider the situation illustrated in Fig.~\ref{fig:yariv_pepper}, where two pump waves counterpropagate through a nonlinear medium at an angle to the $z$-axis and a second pair of waves counterpropagate along the $z$-axis. The fields are taken to be degenerate plane waves with frequency $\omega$, and can be represented by
\begin{align}
  \widetilde E_i(\mathbf{r},t)&=E_i(\mathbf{r})e^{i\omega t}+\mathrm{c.c.}\\
  &=A_i(\mathbf{r})e^{i(\mathbf{k_i}\cdot\mathbf{r}-\omega t)}+\mathrm{c.c.},
\end{align}
for $i=1,2,3,4$, where $A_i(\mathbf{r})$ describes the slowly varying envelope for each wave. With counterpropagating waves, we have
\begin{equation}
  \label{eqn:bfwm_phase_match}
  \mathbf{k_1}+\mathbf{k_2}=0,\quad \mathbf{k_3}+\mathbf{k_4}=0.
\end{equation}

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/yariv_pepper.pdf}
  \end{center}
  \caption[Geometry of backward four wave mixing]{Four wave mixing in the phase conjugate geometry. Two strong pump waves with amplitudes $A_1$ and $A_2$ counterpropagate through a nonlinear medium and at an angle to the $z$-axis. An incident wave propagating along $z$ with amplitude $A_3$ undergoes phase conjugate reflection and a fourth wave with amplitude $A_4$ exits the medium propagating along $-z$.}
  \label{fig:yariv_pepper}
\end{figure}

If the signal and conjugate fields remain weak, it is reasonable to ignore the modification of the pump waves due to the nonlinear interaction. This model also ignores the effects of weak-wave retardation. Furthermore, by considering only the waves indicated in Fig.~\ref{fig:yariv_pepper}, the forward four-wave mixing is also ignored. Phase matching, which enters the analysis for forward four wave mixing, is automatic in the backward geometry. The expression given in Eq.~(\ref{eqn:bfwm_phase_match}), indicates that BFWM is automatically phase matched. These assumptions lead to the following four expressions for the nonlinear polarization produced by the fields in the medium
\begin{equation}
  \label{eqn:polarizations}
  P_1=P_2=0,\quad P_3=6\epsilon_0\chi^{(3)}E_1E_2E_4^*,\quad P_4=6\epsilon_0\chi^{(3)}E_1E_2E_3^*.
\end{equation}
Each of the interacting waves obeys the wave equation in the form
\begin{equation} 
  \label{eqn:wave_eqn} \nabla^2\widetilde E_i-\frac{n^2}{c^2}\frac{\partial^2\widetilde E_i}{\partial t^2}=\frac{1}{\epsilon_0 c^2}\frac{\partial^2\widetilde P_i}{\partial t^2},
\end{equation}
where $n$ is the linear index of refraction in the medium. We see that the nonlinear polarization $\widetilde P_i$ acts as a source term driving the wave equation. Physically, this indicates the role of the polarization in generating new waves. In general, the generated waves may have wavevectors that are not present in the incident fields. It is clear that each term $P_i$ of the polarization given in Eq.~(\ref{eqn:polarizations}) drives the wave equation and gives rise to waves that couple to $E_i$. Thus, the nonlinear polarization of the medium plays the role of coupling the four interacting waves. This coupling can be determined by solving Eq.~(\ref{eqn:wave_eqn}) in the slowly varying amplitude approximation, and neglecting the depletion of the pumps. The result is a pair of amplitude equations for the signal and conjugate waves \cite{Yariv_1977aa}
\begin{align}
  \frac{dA_3}{dz}&=i\kappa A_4^*\\
  \frac{dA_4}{dz}&=-i\kappa A_3^*,
\end{align}
where
\begin{equation}
  \kappa=\frac{3\omega}{nc}\chi^{(3)}A_1A_2,
\end{equation}
is the coupling coefficient.

A physically interesting case is that of a single input $A_3(0)$ at $z=0$, and $A_4(L)=0$. This corresponds to injecting both pump waves, and the signal wave, in order to observe the output spontaneously generated as the fourth wave, which is typically called the conjugate wave for this geometry. In this case the reflected (conjugate) wave at the input ($z=0$) is
\begin{equation}
  \label{eqn:pco_reflect}
  A_4(0)=i\left(\frac{\kappa}{|\kappa|}\tan |\kappa|L\right)A_3^*(0),
\end{equation}
and the transmitted wave at $z=L$ is
\begin{equation}
  \label{eqn:pco_transmit}
  A_3^*(L)=\frac{A_3^*(0)}{\cos|\kappa|L}.
\end{equation}
The most significant implication of this result is that the gain for the transmitted wave given by Eq.~(\ref{eqn:pco_transmit}) is infinite when $|\kappa|L=\pi/2$. The same is true for the reflected wave given by Eq.~(\ref{eqn:pco_reflect}). Physically, this implies that the presence of even a single photon in the mode corresponding to $A_3(0)$ is sufficient to generate a macroscopic field at $A_3(L)$. This infinite gain (and reflectivity) is one origin off-axis beams generated by four-wave mixing processes. The spontaneous generation of this off-axis beam is known as mirrorless parametric oscillation. This analysis has only allowed for the possibility of four plane waves interacting in the system, and conditions exist whereby quantum fluctuations are sufficient to induce new beams of light. The next section introduces another wave-mixing process known as forward four-wave mixing (FFWM). Later sections show that including FFWM in the model of a counterpropagating beam system serves to reduce the gain necessary for self-oscillation and determines the angle at which oscillation occurs.

% section backward_four_wave_mixing (end)

\section{Forward four-wave mixing} % (fold)
\label{sec:forward_four_wave_mixing}

The treatment of Yariv and Pepper specified only enough fields to allow for the possibility of backward four-wave mixing. However, there is another four-wave mixing process with a different geometry that plays a role in generating off-axis fields. Known as forward four-wave mixing, this process takes place between three fields that propagate in the forward direction and are mutually coupled, giving rise to gain in the off-axis beams. The forward four-wave mixing process is illustrated in Fig.~\ref{fig:ffwm}.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/forward_four_wave_mixing.pdf}
  \end{center}
  \caption[Forward four-wave mixing geometry]{Forward four-wave mixing geometry. A strong pump wave with amplitude $A_1$ propagates through a nonlinear medium along the $z$-axis. Two other incident waves, typically taken to be weak probe waves, propagate at an angle to the $z$-axis with amplitudes $A_2$ and $A_3$.}
  \label{fig:ffwm}
\end{figure}

The parametric FFWM process is illustrated in Fig.~\ref{fig:ffwm} where a strong pump wave with amplitude $A_1$ propagates along the $z$-axis and a probe wave with amplitude $A_2$ is incident on the nonlinear medium at a small angle $\theta$. The dashed arrow indicates a second probe wave, also incident at a symmetric angle $\theta$ to the pump wave. To account for FFWM in theoretical treatments, this probe wave must be included initially, but can later be set to have zero incident amplitude. Forward four-wave mixing is the interaction between $A_1$ and $A_2$ that gives rise to gain in the direction of $A_3$. In the grating picture, the pump and probe wave interfere to induce the grating indicated in Fig.~\ref{fig:ffwm}. The pump wave can then scatter off this grating into the direction of $A_3$. 

Treating the coupling between the fields shown in Fig.~\ref{fig:ffwm} begins with an expression for the nonlinear polarization analogous to that given in Eq.~(\ref{eqn:polarizations}). Phase matching is not automatic as it was for the backward geometry; hence, an additional term describes the phase mis-match and  the polarization that gives rise to $A_3$ is given by
\begin{equation}
 \label{eqn:ffwm_polarization} P_3=6\epsilon_0\chi^{(3)}E_1E_1E_3^*\exp\left[i\left(\vec{k}_1+\vec{k}_1-\vec{k}_3\right)\cdot\vec{r}-i\omega t\right].
\end{equation}
Just as in BFWM, each of the interacting waves shown in Fig.~\ref{fig:ffwm} obeys the wave equation given in Eq.~(\ref{eqn:wave_eqn}). The coupling of the three waves is determined by solving (\ref{eqn:wave_eqn}) in the slowly-varying amplitude approximation, and neglecting the depletion of the pumps. The result if a pair of amplitude equations for the two weak waves $A_2$ and $A_3$
\begin{align}
  \label{eqn:ffwm}
  \frac{dA_3}{dz}&=i\kappa A_2^*\exp(i\Delta k z)\\
  \frac{dA_2}{dz}&=i\kappa A_3^*\exp(i\Delta k z),
\end{align}
where $\Delta k=(2\vec{k}_1-\vec{k}_2-\vec{k}_3)\cdot\hat{z}$ is the component of the wavevector mis-match along the $z$-axis, and
\begin{equation}
  \kappa=\frac{3\omega}{nc}\chithree A_1^2,
\end{equation}
is the FFWM coupling coefficient.

Given the waves illustrated in Fig.~\ref{fig:ffwm}, a physically interesting case is that of only one incident off-axis wave, \emph{i.e.}, $A_3(0)=0$ and only the forward pump $A_1(0)$ and $A_2(0)$ are nonzero. The solution to Eqs.~(\ref{eqn:ffwm}), subject to these boundary conditions, is given by
\begin{align}
  \label{eqn:ffwm_solution}
  A_3(z)&=e^{i\Delta k z}\frac{i\kappa}{g}A_2^*(0)\sinh(gz)\\
  A_2(z)&=e^{i\Delta k z}A_2(0)\left[\cosh(gz)-\frac{i\Delta k}{2g}\sinh(gz)\right],
\end{align}
where
\begin{equation}
  g=\sqrt{|\kappa|^2-\left(\frac{\Delta k}{2}\right)^2},
\end{equation}
is a coefficient that describes the gain experienced by both off-axis waves.

The behavior of the amplitude equations give in Eqs.~(\ref{eqn:ffwm_solution}) is not as obvious as it was in the case of the BWFM amplitude equations (Eqs.~(\ref{eqn:pco_reflect}) and (\ref{eqn:pco_transmit})). For this reason, it is useful to consider a specific case and plot the amplitude as a function of position $z$. The three parameters to consider are the initial amplitude of the weak injected wave $A_2(0)$, the coupling parameter $\kappa$, and the phase mismatch $\Delta k$. I have shown in Sec.~\ref{sec:weak_wave_retardation} that weak-wave retardation can serve to phase-match the forward four-wave mixing process at a specific angle $\theta$ between the pump-beam and the off-axis beams. For the purpose of illustration, I will assume this is the case and that the forward four-wave mixing process is phase-matched so that $\Delta k=0$. An appropriate value for the nonlinear coupling can be determined by setting $\kappa L=\pi/4$. This is half the value of $\kappa L$ required for self-oscillation in the Yariv and Pepper treatment, but as the following sections will show, $\kappa L=\pi/4$ is near the threshold condition when both FFWM and BFWM processes are considered simultaneously.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/ffwm_plot.pdf}
  \end{center}
  \caption[Forward four-wave mixing gives rise to gain in off-axis waves.]{The off-axis amplitudes for forward four-wave mixing. The weak injected beam with amplitude $A_2(z)$ (upper curve) experiences gain, and a symmetric off-axis wave with amplitude $A_3(z)$ (lower curve) is generated and grows as it propagates in the medium. The amplitudes are given by Eq.~\ref{eqn:ffwm_solution} with $\Delta k=0$, $A_2(0)=1$ and $A_3(0)=0$.}
  \label{fig:ffwm_plot}
\end{figure}

Figure~\ref{fig:ffwm_plot} illustrates the gain experienced by both off-axis waves $A_2$ and $A_3$. The initial amplitude $A_2(0)=1$ experiences gain as it propagates through the medium. Additionally, the other off-axis wave, with amplitude $A_3(0)=0$, is generated by the interaction and grows with increasing $z$ as well (lower curve in Fig.~\ref{fig:ffwm_plot}). I have assumed the FFWM process is phase matched in this example, but if this is not the case, \emph{i.e.}, if the injected wave is propagating at an angle larger than $\theta\simeq\sqrt{\lambda/2L}$ then the coefficient $g$ decreases for increasing $\Delta k$.

The processes I have considered thus far, BFWM and FFWM, can easily occur simultaneously for beams that propagate with small $\theta$ such that the FFWM process can be phase-matched or nearly phase-matched. By considering the combination of the FFWM process and the BFWM process, Grynberg extends the analysis of Yariv and Pepper \cite{Grynberg_1988ab,Grynberg_1989aa} as described in the following section.

\subsection{Coherent addition of FFWM and BFWM} % (fold)
\label{sec:coherent_addition_of_ffwm_and_bfwm}

In a straightforward extension of the work of Yariv and Pepper, Grynberg demonstrated that including a second probe wave allows both forward and backward four-wave mixing processes to contribute to mirrorless oscillation \cite{Grynberg_1988ab}. The treatment in \cite{Grynberg_1988ab} is a more rigorous version of the arguments I presented above based on weak-wave retardation. Grynberg shows that using the established expressions for the weak- and strong-beam susceptibilities, rather than my estimated phase shift of $\pi/2$, weak-wave retardation causes the forward four-wave mixing process to be phased matched at a specific angle. The result, however, is very similar to my estimate and predicts $\theta$ on the order of a few miliradians depending on specific system parameters.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/grynberg_paye.pdf}
  \end{center}
  \caption[Forward- and Backward-four-wave-mixing add coherently to determine $\theta$.]{Forward- and Backward-four-wave-mixing add coherently to determine $\theta$. The theoretical treatment of both FFWM, and BFWM requires the consideration of six waves. Two counterpropagating pump waves, $E_p$ and $E'_p$, and two pairs of counterpropagating probe waves, $E_1$ ($E'_1$) and $E_2$ ($E'_2$). $\theta$ is the angle between the probe waves and the pump waves.}
  \label{fig:grynberg_paye}
\end{figure}

In a more quantitatively complete treatment, Grynberg and Paye include FFWM, BFWM, and a third parametric wave-mixing processes that occurs in the geometry shown in Fig.~\ref{fig:grynberg_paye} \cite{Grynberg_1989aa}. By considering six waves, this treatment explicitly allows both forward and backward wave mixing processes to occur. Furthermore, the stability of the amplitude equations derived for the four probe waves can be evaluated as a function of the angle $\theta$. Such a stability analysis shows several key features. First, the threshold for mirrorless self-oscillation is infinite for $\theta=0$ implying that no oscillation is possible on the pump beam axis. This is consistent with Silberberg and Bar-Joseph because the fields treated here are degenerate,  whereas Silberberg and Bar-Joseph considered coupling between non-degenerate fields, and predicted no instability for the degenerate case \cite{Silberberg_1982aa}. Second, the lowest threshold occurs for a unique  value of $\theta$ and corresponds to $|\kappa| L=\pi/4$, a factor of 2 smaller than the threshold predicted by Yariv and Pepper. For large $\theta$, the threshold tends toward $|\kappa| L=\pi/2$, the value predicted by Yariv and Pepper, implying that forward four-wave mixing is not important to the analysis for large $\theta$. For large $\theta$, only backward four-wave mixing can be phase matched.

Not only is the instability threshold lower in the model that includes both FFWM and BFWM, but the phase-matching condition for the FFWM process specifies a unique angle $\theta$ where BFWM and FFWM coherently add, thus implying that mirrorless parametric oscillation occurs with conical emission \cite{Grynberg_1989aa}.

% subsection coherent_addition_of_ffwm_and_bfwm (end)

% section forward_four_wave_mixing (end)

\section{Transverse patterns} % (fold)
\label{sec:transverse_patterns}

The treatments of counterpropagating waves presented above have assumed only plane-wave optical fields with distinct wave vectors. Firth and Par\'e extend these treatments by deriving amplitude equations that include the transverse components of the field for counterpropagating fields in a nonlinear medium. By including the transverse degrees of freedom and allowing for multiple four-wave-mixing interactions, they show that transverse effects and counterpropagating pump beams are sufficient ingredients for the system to exhibit spatio-temporal instability \cite{Firth_1988aa}. Furthermore, with the inclusion of transverse components, the threshold for self-oscillation can be significantly lower than that originally shown by Yariv and Pepper, and comparable to that shown by Grynberg and Paye. 

In fact, there are many similarities between the analyses of Grynberg and Paye and Firth and Par\'e. Both models include FFWM, BFWM, and other parametric interactions. Rather than assuming a pair of probe waves at angle $\theta$ to the pump-beam axis, as in \cite{Grynberg_1989aa}, Firth and Par\'e treat one longitudinal dimension, and one continuous transverse dimension with off-axis effects included by analyzing the stability of the wave equation to perturbations with a general wavevector $K$. This treatment can thus be easily generalized to two transverse dimensions. For this reason, it serves as the basis for my numerical simulations presented in Chapter~\ref{cha:nummodel}.

\subsection{Model of Firth and Par\'e} % (fold)
\label{sub:model_of_firth_and_pare}

The interaction considered in this section is familiar from earlier sections, and illustrated in Fig.~\ref{fig:firth_pare}. To study the origin of transverse pattern formation, we consider waves that satisfy Maxwell's equations under the following simplifying assumptions. If the nonlinear medium is isotropic, dispersionless, and the waves propagate primarily along one axis, the wave equation can again be taken to have the form shown in Eq.~(\ref{eqn:wave_eqn}). Consider two waves that counterpropagate through a nonlinear medium and are described by
\begin{align}
  \label{eqn:fields}
  \widetilde E_F\rt&=E_1\rt e^{-i\omega t} +\cc\nonumber\\
  &= F\rt e^{i(kz-\omega t)}+\cc\\
  \widetilde E_B\rt&=E_2\rt e^{-i\omega t} +\cc\nonumber\\
  &= B\rt e^{i(-kz-\omega t)}+\cc,
\end{align}
where $k=n_0\omega/c$, $E_i$ is a field amplitude with spatial dependence $e^{ikz}$, and $F (B)$ is a slowly varying amplitude describing the forward (backward) wave.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/firth_pare.pdf}
  \end{center}
  \caption[Counterpropagating beams in a nonlinear medium]{Two waves with amplitudes $F(\mathbf{r},t)$ and $B(\mathbf{r},t)$ counterpropagate within a nonlinear medium.}
  \label{fig:firth_pare}
\end{figure}

To go from the wave equation to a quantitative description of the field amplitudes in a counterpropagating beam system, we substitute Eq.~(\ref{eqn:fields}) into Eq.~(\ref{eqn:wave_eqn}) and match terms that exhibit the same spatial dependence as the pump beams, thus coupling the waves generated by the nonlinear polarization to the pump waves. The primary difference between this analysis and that of Grynberg and Paye is that the latter considers only two specific probe-beam wavevectors, and here all transverse wavevectors are allowed in the model, and treated as a Fourier decomposition.

There are many possible terms that contribute to the polarization $\widetilde P_i$, but most of them do not couple strongly to either of the pump waves and are thus neglected. The two terms that provide coupling between the pump waves are
\begin{align}
  \label{eqn:firth_p_terms}
  P_1&=3\epsilon_0\chithree\left(E_1^2E_1+2E_1E_2E_2^*\right)\\
  P_2&=3\epsilon_0\chithree\left(E_2^2E_2+2E_2E_1E_1^*\right),
\end{align}
where $P_1$ ($P_2$) generates a wave that couples to $E_1$ ($E_2$). Using the expressions for the incident waves Eq.~(\ref{eqn:fields}), these polarization terms, when substituted into Eq.~(\ref{eqn:wave_eqn}) yield amplitude equations for the forward and backward fields. In doing so, it is usually permissible to make the slowly-varying amplitude (SVA) approximation, which is valid as long as
\begin{equation}
  \left|\frac{d^2E_{1,2}}{dz^2}\right|<<\left|k_{1,2}\frac{dE_{1,2}}{dz}\right|.
\end{equation}
The SVA approximation, which applies to the present situation of continous-wave counterpropagating fields, implies that the fractional change in the field amplitude over a single wavelength is much smaller than unity \cite{Boyd_2002aa}. Under the SVA approximation, the forward and backward fields are described by the following amplitude equations
\begin{align}
  \label{eqn:field_amps}
  \frac{\partial F}{\partial z}+\frac{n_0}{c}\frac{\partial F}{\partial t}-\frac{i}{2k}\frac{\partial^2 F}{\partial x^2}&=i\left[|F|^2+2|B|^2\right]F,\\
  -\frac{\partial B}{\partial z}+\frac{n_0}{c}\frac{\partial B}{\partial t}-\frac{i}{2k}\frac{\partial^2 B}{\partial x^2}&=i\left[2|F|^2+|B|^2\right]B,
\end{align}
where the nonlinear constant $n_2$ has been scaled into the field amplitudes by the relation
\begin{equation}
  F= \sqrt{2 \epsilon_0 n_0 n_2 \omega}F',
\end{equation}
where $F'$ is the field amplitude with physical units. This also has the consequence of letting $I=|F|^2=\kappa$ which allows comparison to the results of the previous section.
It should also be noted that weak-wave retardation is evident in these equations. The term 
$$\left[|F|^2+2|B|^2\right]F$$
shows that the effect of $B$ on $F$ is twice as large as the effect of $F$ on itself. A symmetric term exists in the equations for $B$.

% subsection transverse_model (end)

\subsection{Linear Stability Analysis} % (fold)
\label{sub:linear_stability_analysis}

To determine the stability of the field amplitudes Eqs.~(\ref{eqn:field_amps}), we consider perturbations about the steady-state plane-wave solutions $F_0(z)$ and $B_0(z)$ of the form
\begin{equation}
  \label{eqn:perturbation}
  F(x,z,t)=F_0(z)\left[1+\epsilon f_+(x,z)e^{\lambda t} + \epsilon f_-^*(x,z)e^{\lambda^* t} \right],
\end{equation}
and a similar equation for $B(x,z,t)$. The form of these perturbations (\emph{i.e.}, the inclusion of the conjugate terms) is necessary as the interaction, Eq.~(\ref{eqn:firth_p_terms}), mixes conjugate terms. The perturbations are assumed to obey
\begin{equation}
  \frac{\partial^2}{\partial x^2}f_\pm\simeq -K^2f_\pm,\quad
  \frac{\partial^2}{\partial x^2}b_\pm\simeq -K^2b_\pm,
\end{equation}
where $K$ describes the transverse component of the perturbation. This form of the perturbations allows for a variety of four-wave-mixing couplings, including those treated by Grynberg and Paye \cite{Grynberg_1988ab,Grynberg_1989aa,Firth_1988aa,Firth_1990ab}.
Substituting Eq.~(\ref{eqn:perturbation}) into Eqs.~(\ref{eqn:field_amps}) and linearizing about the steady state solutions gives a set of four coupled equations for the perturbation amplitudes $f_\pm$ and $b_\pm$. Instability will occur whenever these four equations have a nontrivial solution with $\mathrm{Re}\,\lambda>0$. The boundary between stable states and unstable states is known as the instability threshold. Above threshold the system is unstable and perturbations grow, while below threshold the system is stable and perturbations die out and the system returns to its steady state.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/k_squared.pdf}
  \end{center}
  \caption[Phase lag of off-axis waves.]{The off-axis wave with transverse wavevector $K$ and total wavevector $k_w$ propagates at an angle $\theta$ to the pump beam with wavevector $k$.}
  \label{fig:k_squared}
\end{figure}

For this instability, the threshold curve is shown in Fig.~\ref{fig:threshold_intensity}. This plot can be understood to indicate the boundary between stable and unstable states of the system. The vertical axis represents increasing intensity or increasing length. The nonlinear refractive index $n_2$ has been scaled into the field amplitudes so in this model, $IL$ has the same units as $\kappa L$ in the previous sections. The horizontal axis indicates the transverse component of the perturbing wavevector, and the quantity $K^2L/2k$ represents the phase difference between the off-axis waves and the pump waves. This can be seen by considering the pump-beam and off-axis wave with wavevectors illustrated in Fig.~\ref{fig:k_squared}. The difference $\Delta k=k_w-k$ is given by
\begin{equation}
  \Delta k=k_w-k=k\left(\frac{k_w - k}{k}\right)=k\left(\frac{1}{\cos\theta}-1\right)\simeq k \frac{\theta^2}{2}=\frac{K^2}{2k},
\end{equation}
where I have made the small angle approximations, $\theta=K/k$, and $\cos\theta\simeq1-\theta^2/2$. Thus, from Fig.~\ref{fig:threshold_intensity}, we see that, as the intensity increases (imagine moving a horizontal bar up from the $IL=0$ axis), the first point we reach on the curve indicates that the transverse wavevector with the lowest threshold for self-oscillation corresponds to $K^2L/2k\simeq3$, with $IL\simeq 0.45$. Other wavevectors have higher self-oscillation thresholds. The scalloped nature of the threshold curve results from plotting the minima of multiple solutions. In this case, six solutions are shown, each contributing one scallop. These multiple solutions indicate the periodic nature of the phase-matching condition for the underlying four-wave mixing processes. The threshold is lowest for off-axis waves, which are most nearly phase-matched, but there are other off-axis waves that are also nearly phase matched and thus lead to local minima in the instability threshold.

It is also important to note that the on-axis plane-wave ($K=0$) threshold is infinite, in agreement with the results of Silberberg and Bar-Joseph for the case of an instantaneous nonlinear response. Also plotted in Fig.~\ref{fig:threshold_intensity} is the threshold predicted by including only backward four-wave mixing, $IL=\pi/4$. Clearly, the transverse instability occurs at a lower overall threshold, but the threshold curve approaches $IL=\pi/4$ in the limit of large $K$. Hence, in the large-$K$ limit (large $\theta$) the contribution due to forward four-wave mixing is small due to poor phase-matching. 

\begin{figure}[htbp]
  \centering
    \includegraphics[height=3in]{Figures/threshold_intensity.pdf}
  \caption[Threshold intensity for self-focusing media with phase grating]{Threshold intensity for self-focusing media with phase grating, from \cite{Firth_1988aa}. The vertical axis is in units of $IL$, and the horizontal axis is in units of $K^2 L/2k$ where $K$ is the off-axis component of the perturbation wavevector.}
  \label{fig:threshold_intensity}
\end{figure}

Finally, it is instructive to apply these results to a physical situation. As an example, consider a rubidium system. The prediction that self-oscillation occurs at $K^2L/2k\simeq3$ implies that the angle between the waves with wavevector $K$ and the pump waves is approximately $\theta\simeq K/k\simeq3.9$~mrad for a 5~cm sample of rubidium vapor, $\lambda=$780~nm. Furthermore, this result can be compared to the estimate presented in Sec.~\ref{sec:weak_wave_retardation} by writing $\theta$ in terms of $\lambda$ and $L$. Using the small-angle approximation, we have
\begin{equation}
\theta\simeq\frac{K}{k}=\frac{1}{k}\sqrt{\frac{6k}{L}}=\sqrt{\frac{3\lambda}{\pi L}},
\end{equation}
which differs from the weak-wave retardation estimate, $\theta\simeq\sqrt{\frac{\lambda}{2 L}}$, by a factor of $\sqrt{6/\pi}\simeq1.4$.
In the next chapter, I present a pattern forming system consisting of two pump waves that counterpropagate in rubidium vapor. Based on the analysis here, it should not be surprising that this system generates a cone of light that propagates at an angle to the pump waves of $\simeq4$~mrad.

% subsection linear_stability_analysis (end)

% section transverse_patterns (end)

\section{Hexagonal pattern formation} % (fold)
\label{sec:hexagonal_pattern_formation}

Simulations based on the model of Firth and Par\'e exhibit hexagonal pattern formation, which is in agreement with a number of experimental results \cite{Grynberg_1988aa,Pender_1990aa,Tan-no_1980aa,DAlessandro_1991aa,Honda_1995aa}. Grynberg, in \cite{Grynberg_1988ab}, gives a qualitative explanation of the origin of these hexagons. First, recall that the gain of the weak fields in the parametric processes of FFWM and BFWM is associated with the depletion of the pump fields, with two photons being absorbed from the pump beam(s). There are, however, secondary four-wave mixing processes whereby only one pump photon is absorbed and the second input photon comes from the generated off-axis beams. For example, in the forward direction, one pump photon can be absorbed along with one photon from the off-axis beam $E_2$, resulting in emission of two photons, one in beam $E_4$ and one in beam $E_5$  (see Fig.~\ref{fig:grynberg_hexagons}). The phase matching condition requires that the directions of $E_1$, $E_4$, and $E_5$ form an equilateral triangle. Likewise, a pump photon and a photon from $E_1$ can be absorbed, leading to gain in $E_3$ and $E_6$. One can think of this process in terms of \emph{photon generations} where each photon from the first generation can contribute, along with a pump photon, to the creation of a second generation of photons.

\begin{figure}[htbp]
  \begin{center}
    \includegraphics[scale=1]{Figures/grynberg_hexagons.pdf}
  \end{center}
  \caption[Hexagonal patterns in terms of secondary wave-mixing processes]{Hexagonal patterns in terms of secondary wave-mixing processes. a) The six weak waves that form a hexagon, contribute to processes that absorb one pump photon and one weak-wave photon and emit two weak-wave photons. b) These processes happen in both directions, generating hexagons in both outputs.}
  \label{fig:grynberg_hexagons}
\end{figure}

Considering these secondary processes, we count the processes that contribute gain to the weak fields, and those that require absorption from the weak fields. To this order, where one pump photon is absorbed and one off-axis photon is absorbed, there are two forward processes that create a photon for each spot on the hexagon, and only one forward process that requires absorption from that spot. The net result is that a process that breaks the symmetry of the system will cause the pattern to collapse from a Uring pattern to six spots in a hexagon. This symmetry-breaking can be intentional, via misalignment of the pump beams, or unintentional, via aberrations in the windows of a vapor cell, or other optical imperfections. For the case of strongly-broken symmetry, \emph{i.e.} if the pump beams are significantly misaligned, only two spots will be phase matched and the pattern is oriented perpendicular to the plane containing the pump beams.

% section hexagonal_pattern_formation (end)

\section{Higher-order patterns} % (fold)
\label{sec:higher_order_patterns}

The previous section presents a conceptual argument for why counterpropagating plane-waves exhibit hexagonal symmetry in off-axis pattern formation. There are extensions to this argument that explain the generation of patterns with more than six spots. An analysis by Grynberg, Ma\^itre and Petrossian \cite{Grynberg_1994aa} of the single-mirror feedback system, flowerlike patterns are expected when the material exhibits a saturable nonlinearity. In contrast to a Kerr media, where the nonlinear effect is proportional to the intensity without bound, the strength of a saturable nonlinearity increases more slowly above a saturation intensity. Using Gaussian pump beams in a single-mirror feedback geometry propagating through warm rubidium vapor, Grynberg \etal show that patterns with 10 to 24 spots can be generated. Theoretical description of these patterns can be made by considering the conditions under which a Laguerre-Gauss mode with the same waist and phase curvature as the pump beam can oscillate between the nonlinear material and the feedback mirror. The resulting symmetry, \emph{i.e.}, the number of spots, of the driven modes depends on the degree to which the nonlinearity is saturated. For an unsaturated nonlinear response, the pump beams do not couple strongly to higher-order modes and the generated patterns are expected to have few spots. For a saturated nonlinear response, the spatial extent of the nonlinear effect is broadened compared to the Gaussian pump intensity and thus higher-order modes can oscillate, generating more spots.

Although the single-mirror feedback system is distinctly different from the case of two counterpropagating beams, the similarities in observed patterns suggest a similar argument holds for the counterpropagating system as well. As I will show in the next Chapter, my experimental observations are also consistent with this discussion. For pump beam intensities just above threshold, I observe two- and six-spot patterns, and for pump beam intensities well-above threshold, where the nonlinearity is likely to be saturated, I observe similar flowerlike patterns.

% section higher_order_patterns (end)

\section{Polarization instabilities} % (fold)
\label{sec:polarization_instabilities}
The prior discussion of pattern forming instabilities has been based on a scalar model of the nonlinear optical response. In a scalar model, the polarization degree of freedom is explicitly ignored. Polarization instabilities, however, are readily observed in a wide range of counterpropagating-beam systems \cite{Gauthier_1988aa,Gaeta_1993aa,Aumann_2004aa,Ackemann_2001ab}. To discuss the origins of polarization instabilities, I first describe a vector model of the nonlinear optical response. For an isotropic nonlinear material, such as atomic vapor, one can often describe the nonlinear polarization in terms of an effective linear susceptibility
\begin{equation}
  P_i=\epsilon_0\sum_j \chi_{ij}^{\mathrm{eff}} E_j,
\end{equation}
where the indices $i,j=x,y,z$ correspond to the three cartesian coordinates and \cite{Boyd_2002aa}
\begin{equation}
  \label{eqn:chi_eff}
  \chi_{ij}^{\mathrm{eff}} = \left(A-\frac{B}{2}\right) \left(\mathbf{E\cdot E}^*\right)\delta_{ij}+\frac{B}{2}\left(E_i E_j^*+E_i^* E_j\right).
\end{equation}
The second term in Eq.~(\ref{eqn:chi_eff}) allows a field polarized in one direction to drive a material polarization that will radiate with a different state of polarization. As an example, one of the terms contained in Eq.~(\ref{eqn:chi_eff}) gives rise to a nonlinear polarization of the form
\begin{equation}
  P_x=\epsilon_0 \frac{B}{2}\left(E_x E_y^*+E_x^* E_y\right)E_y,
\end{equation}
which describes the coupling between two orthogonally polarized fields $E_x$ and $E_y$ via polarization $P_x$. Clearly, this model requires $B\neq0$ in order for the two polarization states to couple, \emph{i.e.}, if $B=0$ no polarization instability is expected.

The relative magnitudes of the $A$ and $B$ coefficients in this model depend on the nature of the physical processes responsible for the optical nonlinearity \cite{Boyd_2002aa}. For optical nonlinearity arising from resonant atomic response, the $A$ and $B$ coefficients are determined by the different angular momenta of the atomic states involved \cite{Saikan_1982aa}.

Gaeta and Boyd \cite{Gaeta_1993aa} perform a linear stability analysis of counterpropagating beams that extends the analysis of Firth and Par\'e by including the vector nature of the field, \emph{i.e.}, allowing polarization degrees of freedom. This analysis is based on the effective-$\chi$ model presented above [Eq.~(\ref{eqn:chi_eff})] and shows that the nature of the instability with the lowest threshold depends both on the ratio $B/A$ and on the polarization states of the counterpropagating pump beams. My experiment utilizes linearly co-polarized pump beams, a configuration that Gaeta and Boyd show is unstable to off-axis amplitude fluctuations, as in the Firth and Par\'e treatment. However, as long as $B/A$ is positive, this configuration does exhibit a polarization instability at input intensities that are higher than the intensity needed for amplitude instabilities.

The polarization instabilities studied in \cite{Gaeta_1993aa} are predicted by the treatment based on Eq.~(\ref{eqn:chi_eff}). However, Pinard \etal \cite{Pinard_1994aa} have shown that this treatment is not complete for the case of two counterpropagating beams in Doppler-broadened medium. They find that, instead of two constants, $A$ and $B$, five constants are required in general to describe the nonlinearity, but still in the third-order ($\chi^{(3)}$) approximation. For the case of an optical pumping nonlinearity, only four of these are independent and they correspond to the magnetization and electric-quadrupole moment of the medium induced by the forward and backward beams. Naturally, these parameters depend on the angular momentum states coupled by the counterpropagating beams, and thus depend on the specific transitions excited by the pump fields.

To determine how well these treatments agree with my experimental results, I consider the following details. As I will discuss in Sec.~\ref{sub:pump_beam_frequency}, I observe pattern formation on the high-frequency side of the $^5\text{S}_{1/2}(F=1) \rightarrow {^5\text{P}_{3/2}}(F'=1)$ hyperfine transition in rubidium. Based on the treatment given in Ref.~\cite{Saikan_1982aa}, these angular momentum states lead to $B=0$ and hence Eq.~\ref{eqn:chi_eff} predicts that there is no polarization instability due to the $(F=1) \rightarrow (F'=1)$ transition. The treatment by Pinard \etal \cite{Pinard_1994aa} makes a similar prediction: the $(F=1) \rightarrow (F'=1)$ transition does not give rise to polarization instability. Clearly, these treatments alone are not adequate for explaining my observation of a polarization instability near Rb $^5\text{S}_{1/2}(F=1) \rightarrow {^5\text{P}_{3/2}}(F'=1)$.

My observation of off-axis pattern formation for pump beams tuned above the $(F=1) \rightarrow (F'=1)$ transition is, however, consistent with the treatment given by Pinard \etal~\cite{Pinard_1994aa}. The angular momenta of these states are appropriate for observing self-focusing on the high-frequency side of the resonance. Self-focusing, or positive $n_2$, is assumed in the model of Firth and Par\'e and is crucial for off-axis pattern formation as it is required for the forward four-wave mixing process to be phase-matched (see Sec.~\ref{sec:weak_wave_retardation}).

The poor agreement between these treatments and my observation of polarization instability suggests that either Eq.~(\ref{eqn:chi_eff}) is not adequate to describe the nonlinearity in my system, or the results of Saikan \cite{Saikan_1982aa} are not accurate for my case. Pinard \etal suggest the former is true and generalize Eq.~(\ref{eqn:chi_eff}) to treat the case of counterpropagating beams in a Doppler-broadened medium. However, their treatment also fails to account for my observation of a polarization instability. Some features of my experiment, such as which side of resonance exhibits self-focusing are correctly described, but further theoretical treatment is necessary to describe the origins of the polarization instability I observe. The perturbative (third-order) approach of Pinard \etal assumes that the two-level transition is unsaturated and that the ground state is closed. I operate well above the two-level saturation intensity and, in rubidium, the $(F'=1)$ excited state can decay to the $(F=2)$ ground state which is detuned from the pump beams by over 6~GHz (\emph{i.e.}, the transition is open). Thus, a theoretical treatment based on $F$-state pumping or on a non-perturbative approach may yield more accurate results.

% section polarization_instabilities (end)


\section{Summary} % (fold)
\label{sec:patterns_summary}

In this Chapter, I have described the forward and backward four-wave mixing processes (FFWM and BFWM), and presented three theoretical treatments that consider the roles these processes play in instabilities that spontaneously generate off-axis light. The treatment of Yariv and Pepper considers only backward four-wave mixing and describes an instability that is automatically phase-matched to occur in any direction \cite{Yariv_1977aa}. Grynberg and Paye refine this treatment by considering the combination of forward and backward four-wave mixing processes and show that the two processes only add coherently at a finite angle $\theta$ leading to conical emission of instability generated light. In another treatment considering both FFWM and BFWM, Firth and Par\'e include a continuous transverse dimension allowing for arbitrary transverse wavevector components. This analysis also predicts that instability-generated light will be emitted at a finite angle $\theta$ on the order of a few mrad. Furthermore, the treatments of Grynberg and Paye, and Firth and Par\'e predict instability thresholds that are roughly half that predicted by Yariv and Pepper \cite{Grynberg_1989aa,Firth_1988aa}.

Additionally, I have given a qualitative argument, originally presented by Grynberg, that suggests conical emission is replaced by hexagonal pattern formation when weak symmetry-breaking is present in the system \cite{Grynberg_1988ab}. I have also argued that higher order patterns can be observed and are expected for systems exhibiting saturable nonlinearities. Finally, I have described polarization instabilities, and several treatments that predict polarization instabilities for various experimental conditions. These treatments do not account for all of my experimental observations, however, they serve to introduce the relevant concepts and suggest directions for future theoretical work. The next Chapter presents my experimental observations of counterpropagating beam instabilities, including pattern formation, in a system where warm rubidium vapor serves as the nonlinear medium.

% section summary (end)





